Aslı Pekcan

2 + 1 KdV(N) equations

We present some nonlinear partial differential equations in 2 + 1-dimensions derived from the KdV equation and its symmetries. We show that all these equations have the same 3-soliton solution structures. The only difference in these solutions are the dispersion relations. We also show that they possess the Painleve property.

On the classification of Darboux integrable chains

We study a differential-difference equation of the form tx(n+1)=f(t(n),t(n+1),tx(n)) with unknown t=t(n,x) depending on x and n. The equation is called a Darboux integrable if there exist functions F (called an x-integral) and I (called an n-integral), both of a finite number of variables x,t(n),t(n±1),t(n±2),…,tx(n),txx(n),…, such that DxF=0 and DI=I, where Dx is the operator of total differentiation with respect to x and D is the shift operator: Dp(n)=p(n+1). The Darboux integrability property is reformulated in terms of characteristic Lie algebras that give an effective tool for classification of integrable equations. The complete list of equations of the form above admitting nontrivial x-integrals is given in the case when the function f is of the special form f(x,y,z)=z+d(x,y).

Complete list of Darboux integrable chains of the form t1x = tx +d (t, t1)

We study differential-difference equation (d/dx)t(n+1,x)=f(t(n,x),t(n+1,x),(d/dx)t(n,x)) with unknown t(n,x) depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, {t(n+k,x)}k=−∞∞, {(dk/dxk)t(n,x)}k=1∞, such that DxF=0 and DI=I, where Dx is the operator of total differentiation with respect to x and D is the shift operator: Dp(n)=p(n+1). Reformulation of Darboux integrability in terms of finiteness of two characteristic Lie algebras gives an effective tool for classification of integrable equations. The complete list of Darboux integrable equations is given in the case when the function f is of the special form f(u,v,w)=w+g(u,v).

SOLUTIONS OF THE EXTENDED KADOMTSEV–PETVIASHVILI–BOUSSINESQ EQUATION BY THE HIROTA DIRECT METHOD

We show that we can apply the Hirota direct method to some non-integrable equations. For this purpose, we consider the extended Kadomtsev–Petviashvili–Boussinesq (eKPBo) equation with M variable which is where aij = aji are constants and xi = (x,t,y,z, …,xM). We will give the results for M = 3 and a detailed work on this equation for M = 4. Then we will generalize the results for any integer M > 4.

Характеристическая алгебра Ли и классификация полудискретных моделей@@@Characteristic Lie algebra and classification of semidiscrete models

Differential-difference equation

On Some Algebraic Properties of Semi-Discrete Hyperbolic Type Equations

\frac{d}{dx}t(n+1,x)=f(x,t(n,x),t(n+1,x),\frac{d}{dx}t(n,x))

Integrable Nonlocal Reductions

with unknown

Solutions of Non-Integrable Equations by the Hirota Direct Method

t(n,x)